Unveiling the impact of population density in SDM:

Response curve and variable importance could deliver misleading inference

Jeon, Cheongok1, Kim, Daehyun2

1The Institute for Korean Regional Studies, Seoul National University, Seoul 08826; LophitaL@snu.ac.kr

2Department of Geography, Seoul National University, Seoul 08826; biogeokim@snu.ac.kr

• Estimating how species react to their environments

• Based on correlation between species occurrence and environment

https://damariszurell.github.io/SDM-Intro/

Species Distribution Model(SDM)?

•Model habitat suitability or occurrence probability

•Fitting better model =

Searching for ”best” line against certain environmental variable

• Virtual species simulation =

researcher draw own line and check if SDM can estimate it

SDM!

https://damariszurell.github.io/EEC-MGC/index.html

Assumption of SDM

• Species Distribution Model(SDM)

• Virtual species simulation

•Response curve in SDM,

•Response function in simulation

Response curve is constant over the species

Assumption of SDM

Response curve could depend

on local population density

• # of population density affect energy savings

• Saved energy could be used for other fitness, such as reproduction rate

Gilbert, et al., J Exp Biol (2008)

Response curve may not constant over the species

How species react to their environments could

depends on their population density

Social

thermoregulation

Antipredator

vigilance

Resistance to

extreme weather event

Allee effect could be accounted

to estimate fitness/habitat suitability

•Allee effect: positive relationship between individual fitness and numbers/density of conspecifics

•More conspecifics, higher survival/reproduction rate

•Less conspecifics, low survival/reproduction rate

•Habitat suitability could be correlated with # of conspeicifics, population density

Stephens, et al. 1999

Individual fitness

Abundance

No Allee effect Strong Allee effect

Knowledge gap

•Population density could indirectly affect habitat suitability by Allee effect

e.g., social thermoregulation(abiotic), antipredator vigilance(biotic)

•Needs for how population density could affect inference of SDM

•Many focus on prediction error, not interpretation error

Research question

•

How to account Allee effect on SDM?

•

How they comes out from different SDM algorithm and interpretation methods?

Methods

•Virtual species simulation

•Equation solving with Monte Carlo method

•SDM with various algorithms(presence-background, presence-absence)

•Model interpretation(response curve, variable importance)

•For response curve-SHAP(SHapley Additive exPlanations), PDP(Partial Dependency Plot)

•For variable importance-mean of Shapley value, permutation variable importance

Simulation settings

Where 𝑥!⋅⋅⋅ 𝑥"∈ 𝑋"

𝑌

#$ =1

σ! %&

𝑒𝑥𝑝 −1

𝑥!− µ!

σ!

+⋅⋅⋅ + 1

σ" %&

𝑒𝑥𝑝 −1

𝑥"− µ"

σ"

•Habitat suitability(𝑌

#$) is determined by environmental varible(𝑋")

with Gaussian curve

σ"

'= 𝑎()(."

'𝑋()( + σ"

•Standard deviation(SD) affected with Allee effect(σ"

')could be linear equation of

population density(𝑋()(), Allee effect coefficient(𝑎()(."

') and SD for isolated situation(σ"

'')

Assume population density is equal to habitat suitability (𝑋()( = 𝑌

#$)

𝑋()( =1

𝑎()(⋅!

'𝑋()( + σ!

'' 2π exp −1

𝑥!− µ!

𝑎()(⋅!

'𝑋()( + σ!

+⋅⋅⋅ + 1

σ"%&

exp −1

𝑥"− µ"

σ"

Results

• Presenting noble method to account Allee effect

•"Bullet-shape" curve from positive Allee effect coefficient(𝑎()(."

•"Triangle-shape" curve from negative Allee effect coefficient(𝑎()(."

Results: response curve

• For response curve, flexible presence-absence algorithms

performed better than presence-background algorithms

• BRT, GAM, was effective, followed by MaxEnt and RFds

• Generalized Linear Model(GLM) showed the poorest performance

• MaxEnt showed highest variance from Allee effect coefficient(a!"!.$

• SHAP showed slightly more accurate interpretation than PDP

• Species with population density effected have higher variance on

RMSE than not effected species

Results: variable importance

• For variable importance, most of model effectively estimated

variable importance when population density is not depended

except GLM.

• However, most of model couldn’t estimate variable importance

when they comes to population density effected species

• This limitation was more serious on SHAP

• Absolute mean of Shapley value was not effective

to capture variable importance under population density

depended response curve

• BRT and GAM was effective on estimating response curve but

not on variable importance

Discussion

• The main reason of performance degradation could be information limitation

• Allee effect could cause spatial clustering of presence-absence data, which could limit the

environmental information with positive spatial autocorrelation of environmental variables.

Presence

Abs. Abs.

Environmental variable

Population density

Discussion

•Our method could be more useful to model ecosystem engineer species and

positive feedback of habitat suitability

• Accounting population density could alter our understanding about habitats

Population density

Habitat suitability

Abiotic

In case of Ecosystem engineer

Increase in habitat modification

Increase habitat suitability

Increase in population

Discussion

• Spatial cluster of presence dataset is often called “bias” but if this implicit density, affecting

their habitat suitability, “bias removed” model interpretation could be far from truth

• Prediction performance are not always equal to interpretation performance,

needs for explanatory modeling analysis

Valavi, et al. 2021

Fourcade, et al. 2021

Conclusion

• Population density depended response curve could hinder proper model interpretation

• GAM could be most robust algorithm for non-linear relationship and

population density depended response curve

• Certain methods could be less accurate to interpret species response,

(Especially, SHAP for variable importance)

• Presence-background methods could be alternative,

but interpretation form various approach is recommended

• Current SDM methods could deliver misleading inference

with species that had strong Allee effect

• Allee effect and its effect on habitat suitability needs more attention

Thank you!

Jeon, Cheongok1and Kim, Daehyun2

1The Institute for Korean Regional Studies, Seoul National University, Seoul 08826; LophitaL@snu.ac.kr

2Department of Geography, Seoul National University, Seoul 08826; biogeokim@snu.ac.kr

@biogeojeon

cheongokjeon@gmail.com

lophital

Reference

1. Stephens, P. A., Sutherland, W. J., & Freckleton, R. P. (1999). What Is the Allee Effect? Oikos, 87(1), 185

2. Gilbert, C.,Blanc, S.,Le Maho, Y.,Ancel, A., 2008, Energy saving processes in huddling emperor penguins: From experiments to theory, Journal of Experimental Biology., 211,

1–8

3. Angulo, E.,Luque, G. M.,Gregory, S. D.,Wenzel, J. W.,Bessa-Gomes, C.,Berec, L.,Courchamp, F., 2018, Allee effects in social species, Journal of Animal Ecology., 87, 47–58

4. Fourcade, Y., Engler, J.O., Rödder, D., & Secondi, J. (2014). Mapping Species Distributions with MAXENT Using a Geographically Biased Sample of Presence Data: A

Performance Assessment of Methods for Correcting Sampling Bias. PLoS ONE, 9.

5. Valavi, R., Guillera-Arroita, G., Lahoz-Monfort, J. J., & Elith, J. (2022). Predictive performance of presence-only species distribution models: a benchmark study with

reproducible code. Ecological Monographs, 92(1), e01486.

Limitation

• Strong assumption about habitat suitability and population density relationship

• Mobility factors should be accounted on further research, especially for simulation methods

• Hyperparameter tuning could alter fitting better model but measuring residuals are limited

• Spatial scale of population density effect is not considered

Residuals-response curve

• For population density effected 6 environmental variable,

Red line indicate MaxEnt SHAP interpretation and

Black line for response function for virtual species

Green dots indicate presence datasets

Residuals-SHAP

• Each dot indicate datasets to

measure residauls on response

curve

• Y axis indicate whether model

interpretation was

over-estimate or

under-estimated